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Section 11.6 Spectra

Freudenthal’s Suspension Theorem suggests packaging the iterated suspensions of a space into a single object that records its stable behavior. This is the notion of a spectrum.

Definition 11.17.

A spectrum \(X_\bullet\) is a sequence of based spaces \(\{X_n\}_{n=0}^\infty\) together with structure maps \(\Sigma X_n \to X_{n+1}\text{.}\)

Example 11.18. Suspension spectrum.

Every based space \(K\) gives rise to its suspension spectrum \(\Sigma^\infty K\text{,}\) consisting of the spaces \(\{\Sigma^n K\}_{n=0}^\infty\) with structure maps \(\Sigma \Sigma^n K \xrightarrow{\;\cong\;} \Sigma^{n+1} K\text{.}\)

Definition 11.19.

The homotopy groups of a spectrum \(X_\bullet\) are
\begin{equation*} \pi_q(X_\bullet) := \operatorname*{colim}_{n} \pi_{q+n}(X_n), \qquad q \in \mathbb{Z}, \end{equation*}
where the colimit is taken along the maps
\begin{equation*} \pi_{q+n}(X_n) \xrightarrow{\;\Sigma\;} \pi_{q+n+1}(\Sigma X_n) \to \pi_{q+n+1}(X_{n+1}). \end{equation*}
Only the terms with \(q + n \geq 2\) matter for the colimit, so \(\pi_q(X_\bullet)\) is an abelian group, and \(q\) may be negative.
By construction, \(\pi_q(\Sigma^\infty K) = \pi_q^{s}(K)\text{.}\) If \(K\) is a CW complex this vanishes for \(q < 0\text{,}\) since \(\Sigma^n K\) is \((n-1)\)-connected. A general spectrum can have nonzero homotopy groups in negative degrees (Exercise 11.25).

Example 11.20. Eilenberg–MacLane spectrum.

We will see in coming chapters that every generalized cohomology theory gives rise to a spectrum. In particular, singular cohomology with coefficients in an abelian group \(A\) is represented by the Eilenberg–MacLane spectrum \(HA\text{,}\) characterized up to homotopy by
\begin{equation*} \pi_q(HA) = \begin{cases} A & \text{if } q = 0, \\ 0 & \text{otherwise}. \end{cases} \end{equation*}
Its \(n\)th space is an Eilenberg–MacLane space \(K(A, n)\text{,}\) a space whose only nontrivial homotopy group is \(\pi_n = A\) (Chapter 13); this gives the formula above.

Definition 11.21.

The suspension spectrum of \(S^0\) is called the sphere spectrum, denoted \(\mathbb{S} := \Sigma^\infty S^0\text{.}\) Its homotopy groups are the stable stems, \(\pi_q(\mathbb{S}) = \pi_q^s\text{.}\)

Why spectra?

There are two complementary reasons.
  1. Stable homotopy groups are more tractable. A foundational result, Serre’s finiteness theorem (which we will prove in Chapter 23, see Corollary 23.8), asserts that
    \begin{equation*} \pi_q(\mathbb{S}) = \pi_q^{s}(S^0) = \begin{cases} \mathbb{Z} & \text{if } q = 0, \\ \text{a finite abelian group} & \text{if } q > 0, \end{cases} \end{equation*}
    and \(\pi_q(\mathbb{S}) = 0\) for \(q < 0\text{.}\) No analogue holds for unstable homotopy groups. For example, \(\pi_3(S^2) \cong \mathbb{Z}\) (Example 11.13) shows that finiteness can fail off the diagonal.
  2. Spectra unify topology and algebra. Every space gives a spectrum (its suspension spectrum), and every abelian group gives a spectrum (\(HA\)). So one can “do algebra with spectra”. The resulting framework of higher or homotopical algebra, sometimes called Waldhausen’s brave new rings paradigm, treats the sphere spectrum \(\mathbb{S}\) as the initial commutative ring, playing the role that \(\mathbb{Z}\) plays in classical algebra.
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